The number of functions $$f$$, from the set $$\mathrm{A}=\left\{x \in \mathbf{N}: x^{2}-10 x+9 \leq 0\right\}$$ to the set $$\mathrm{B}=\left\{\mathrm{n}^{2}: \mathrm{n} \in \mathbf{N}\right\}$$ such that $$f(x) \leq(x-3)^{2}+1$$, for every $$x \in \mathrm{A}$$, is ___________.

Let for the $$9^{\text {th }}$$ term in the binomial expansion of $$(3+6 x)^{\mathrm{n}}$$, in the increasing powers of $$6 x$$, to be the greatest for $$x=\frac{3}{2}$$, the least value of $$\mathrm{n}$$ is $$\mathrm{n}_{0}$$. If $$\mathrm{k}$$ is the ratio of the coefficient of $$x^{6}$$ to the coefficient of $$x^{3}$$, then $$\mathrm{k}+\mathrm{n}_{0}$$ is equal to :

$$ \frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+\cdots+ \frac{30^{3}-29^{3}+28^{3}-27^{3}+\ldots+2^{3}-1^{3}}{15 \times 63}$$ is equal to _____________.

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $$\tan ^{-1} \frac{3}{4}$$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is ______________.